1. Field of the Invention
The present invention relates to and provides with a concrete basis the subspace methods for signal analysis used in direction of arrival (DOA) estimation and in exponentially damped sinusoids (EDS) modeling. The DOA estimation is essential task of radar, sonar and other similar systems, where an array of sensors is used to detect and locate some wave-reflecting or emitting objects, and the EDS modeling has practical application in various areas including the digital audio. In the invention field the samples obtained by analog-to-digital conversion (ADC) of the respective sensors signals are put into a data matrix that is numerically processed through a computer needed to achieve the desired results of the DOA estimation or the EDS modeling in real-time.
2. Description of the Related Art
The popularity of the subspace methods (see A. J. van der Veen et al., “Subspace Based Signal Analysis using Singular Value Decomposition”, September 1993, Proc. IEEE, vol. 81, pp. 1277-1308) is owing to their superior spatial or frequency resolution at feasible signal-to-noise ratios (SNR). These methods consider the space spanned by the eigenvectors of the product XX′ of the processed L×M data matrix X and its complex conjugate transpose X′.
In the DOA estimation the M columns of X are complex-valued ‘snapshots’ across an array of L≧2 sensors, which signals are put in the rows of X, and as a rule M≧100, M>>L.
In the EDS modeling the observed series x=[x0, x1, . . . , xn-1] of n≧100 real or complex-valued signal samples is set into a structured (Hankel) L×M matrix X with L=M−1 or L=M and L+M−1=n. The EDS model of x is
                                          x            i                    =                                                    ∑                                  k                  =                  1                                K                            ⁢                                                c                  k                                ⁢                                  z                  k                  i                                                      +                          w              i                                      ⁢                                  ⁢                              i            =            0                    ,          1          ,          …          ⁢                                          ,                      n            -            1                                              (        1        )            where K is the model order, ck are the amplitudes of the components, zk=exp[(−αk+j2πfk)Δt] are the signal poles with damping factors αk and frequencies fk, j is the imaginary unit, Δt is the sampling interval of x and wi is the random noise in xi. Owing to its generality the EDS model is used in various areas including the digital audio, where an adequate application of this model can originate improved coding schemes.
In the DOA estimation the eigenvectors are obtained mainly by direct computation and eigenvalue decomposition (EVD) of the product XX′, while the subspace methods in the EDS modeling use singular value decomposition (SVD) of the processed data matrix X:X=UΣV′  (2)where U and V are L×L and M×M unitary matrices and E is L×M diagonal matrix containing the singular values σX,1≧σX,2≧ . . . ≧σX,L≧0 of X. From (2) and the EVD definition it follows that the squared singular values of X and the columns of its left singular matrix U represent the eigenvalues and the eigenvectors of the product XX′.
All subspace methods are based on the partition of the space span {U} into a dominant part (referred to as signal subspace) spanned by the first K columns of U and a secondary part (referred to as noise subspace) spanned by the last L−K≧1 columns of U. For data matrices X with finite sizes this partition is plain only in the case of noiseless X henceforth denoted as S, when the singular values are σS,1≧ . . . ≧σS,K>σS,K+1= . . . =σS,L=0 and the subspace methods are exact. Provided that the signal subspace dimension K (that in the DOA estimation represents the number of detected objects and in the EDS modeling represents the model order) is known, most methods perform satisfactorily also in the presence of some additive noise, but the determination of K for noisy data matrices is an open problem.
Usually in the DOA estimation the signal and noise subspaces are partitioned by the information theoretic criteria (ITC), that are based on the properties of the sample covariance matrix XX′/M with M→∞. When the rows of X contain uncorrelated zero mean white noise W of variance σW2, for very large M the squared singular values σX,k2 of X=S+W approach σS,k2+MσW2, where σS,k are the above-cited singular values of the noiseless S and hence the last L-K eigenvalues of XX′ are asymptotically equal to MσW2. Since all ITC presume similar uniformity, they are appropriate only in the cases of high SNR, otherwise tend to overvalue the signal subspace dimension K and cause false alarms in the systems for DOA estimation. Therefore some more elaborated methods for determination of the signal subspace dimension K by using the eigenvalues of the product XX′ are proposed, however generally they are very difficult to tune and automate.
The eigenvalue-based techniques for the subspaces partition perform worse in the EDS modeling, where the signal components are non-stationary and have highly varied amplitudes. Besides the assumption for uncorrelated noise is inconsistent with the Hankel structure of the data matrices X, that in the cases of real-valued signals with pairs of complex conjugate terms in the model (1) contain only L+M−1 instead of 2LM independent measurements used in the DOA estimation. These difficulties are surmounted to some extent by the recent eigenvector-based techniques for EDS model order selection (see R. Badeau et al., “A new perturbation analysis for signal enumeration in rotational invariance techniques”, February 2006, IEEE Trans. SP, vol. 54, pp. 450-458 and J. M. Papy et al., “A Shift Invariance-Based Order-Selection Technique for Exponential Data Modeling”, July 2007, IEEE Signal Processing Letters, vol. 14, pp. 473-476). Negative traits of such techniques are burdensome computations, omission of some weak components if there are much stronger ones, unsteady results at low SNR and impossibility to cover the case of signal subspace dimension K equal to zero, that arises when the processed data matrix X contains only (or highly predominating) noise W.
What the subspace methods used in DOA estimation and in EDS modeling need is a technique for determination of the signal subspace dimension K that is computationally effective and easily tunable, causes very few false alarms, operates accurately within or below the SNR limits attained by others techniques and downwards determines a steadily decreasing to zero dimension K of the signal subspace.